To date, there are several types of OFDM type multi-carrier modulations.
Among these types, the most classic modulation technique comprises a particularly simple equalization system based on the insertion of a guard interval. This guard interval, also called a cyclical prefix, provides for efficient performance in the face of echoes, at the cost of a loss of spectral efficiency.
During this guard interval, no payload information is transmitted. This is to ensure that all the information received comes from a same symbol. Thus, the various phenomena of echoes due to ISI (inter-symbol interference) and to the Doppler effect are efficiently curbed.
OFDM/OQAM (“Orthogonal Frequency Division Multiplexing/Offset Quadrature Amplitude Modulation”) is an alternative to this classic OFDM modulation and has been designed to prevent this loss of spectral efficiency caused by the introduction of a guard interval.
More specifically, OFDM/OQAM does not necessitate the presence of a guard interval or a cyclical prefix because of a judicious choice of the prototype function modulating each of the carriers of the signal. This function must be well localized in the time-frequency space.
It may be recalled indeed that the set of carriers of a multi-carrier modulation forms a multiplex and that each of the carriers of this multiplex is shaped by means of a same prototype function, written as g(t), which characterizes the multi-carrier modulation. With v0 denoting the spacing between two adjacent carriers of the multiplex and τ0 denoting the temporal spacing between two multi-carrier symbols sent, the signal sent, at each instant nτ0, on the mth subband with vm, the center frequency is αm,neiφm,ne2iπvmtg(t−nτ0), where the αm,n values represent the digital data to be transmitted. The expression of the signal sent in baseband (centered on the frequency Mv0) is then:
                              s          ⁡                      (            t            )                          =                              ∑            n                    ⁢                                    ∑                              m                =                0                                                              2                  ⁢                  M                                -                1                                      ⁢                                          a                                  m                  ,                  n                                            ⁢                              ⅇ                                  ⅈφ                                      m                    ,                    n                                                              ⁢                              ⅇ                                  2                  ⁢                  ⅈπ                  ⁢                                                                          ⁢                  m                  ⁢                                                                          ⁢                                      ν                    0                                    ⁢                  t                                            ⁢                              g                ⁡                                  (                                      t                    -                                          n                      ⁢                                                                                          ⁢                                              τ                        0                                                                              )                                                                                        (        1        )            
It will be noted here that, for the sake of clarity, we envisage the case of a signal having an even number of frequency subbands. The signal can of course be written more generally in the form:
                              s          ⁡                      (            t            )                          =                              ∑            n                    ⁢                                    ∑                              m                =                0                                            M                -                1                                      ⁢                                          a                                  m                  ,                  n                                            ⁢                              ⅇ                                  ⅈφ                                      m                    ,                    n                                                              ⁢                              ⅇ                                  2                  ⁢                  ⅈπ                  ⁢                                                                          ⁢                  m                  ⁢                                                                          ⁢                                      ν                    0                                    ⁢                  t                                            ⁢                              g                ⁡                                  (                                      t                    -                                          n                      ⁢                                                                                          ⁢                                              τ                        0                                                                              )                                                                                        (        2        )            
It will indeed be recalled that, according to a classic technique, digital data αm,n of zero value are introduced at the edges of the spectrum, thus modifying the number of terms that effectively play a role in the above sum, and enabling for example the operation to be reduced to an even number of carriers.
The functions gm,n(t)=eiφm,ne2iπmv0tg(t−nτ0) are called the “time-frequency” translates of g(t). To retrieve the information transmitted by each of the carriers, it is necessary to choose g(t) and the phases φm,n so that the above “time-frequency” translates are separable. A sufficient condition for verifying this property of separability is that the translates should be orthogonal in the sense of a scalar product defined on the set of finite energy functions (which is a finite Hilbert space in the mathematical sense).
The space of the finite energy functions accepts the following two scalar products:                the complex scalar product        
      〈          x      ❘      y        〉    =            ∫      R        ⁢                  x        ⁡                  (          t          )                    ⁢                        y          *                ⁡                  (          t          )                    ⁢              ⅆ        t                            the real scalar product        
            〈              x        ❘        y            〉        R    =      ℛ    ⁢                  ⁢    e    ⁢                  ∫        R            ⁢                        x          ⁡                      (            t            )                          ⁢                              y            *                    ⁡                      (            t            )                          ⁢                  ⅆ          t                    
Thus two types of multi-carrier modulation are defined:                a complex type multi-carrier modulation for which the chosen function g(t) guarantees orthogonality, in the complex sense, of its translates. This is the case, for example, with classic OFDM modulation, also called OFDM/QAM (“Orthogonal Frequency Division Multiplexing/Quadrature Amplitude Modulation”). For such a modulation, φm,n=0 and αm,n are complex data.        a real type multi-carrier modulation for which the chosen function g(t) guarantees orthogonality, in the real sense, of its translates. This is the case, for example, with OFDM/OQAM modulations. For a modulation of this type, φm,n=(π/2)*(m+n) and the pieces of data αm,n are real data.        
Thus, a transmitted OFDM/OQAM signal can be written as follows:
                              s          ⁡                      (            t            )                          =                              ∑            n                    ⁢                                    ∑                              m                =                0                                            M                -                1                                      ⁢                                          a                                  m                  ,                  n                                            ⁢                                                                    ⅈ                                          m                      +                      n                                                        ⁢                                      ⅇ                                          2                      ⁢                      ⅈπ                      ⁢                                                                                          ⁢                      m                      ⁢                                                                                          ⁢                                              ν                        0                                            ⁢                      t                                                        ⁢                                      g                    ⁡                                          (                                              t                        -                                                  n                          ⁢                                                                                                          ⁢                                                      τ                            0                                                                                              )                                                                                        ︸                                                            g                                              m                        ,                        n                                                              ⁡                                          (                      t                      )                                                                                                                              (        3        )            where αm,n is the real symbol sent on the mth subcarrier at the nth symbol time, M is the number of carriers, v0 is the intercarrier spacing, τ0 represents the duration of an OFDM/OQAM symbol, and g is the prototype function.
This prototype function g modulating each OFDM/OQAM carrier must be very well localized in the time domain to limit inter-symbol interference. Furthermore, it must be chosen so as to be very well localized in the frequency domain, to limit inter-carrier interference (due to the Doppler effect, phase noise etc. . . . ). This function must also guarantee orthogonality between sub-carriers.
The mathematical functions that show these characteristics exist but the best localized among them ensure orthogonality solely on real values. For this reason, the symbols transmitted by OFDM/OQAM modulation must be with real values so that they can be retrieved without interference at reception.
Orthogonality between the time frequency translates of the prototype function is guaranteed if:
                              Re          (                                    ∫              ℛ                        ⁢                                                                                g                                          m                      ,                      n                                                        ⁡                                      (                    t                    )                                                  ·                                                      g                                                                  m                        ′                                            ,                                              n                        ′                                                              *                                    ⁡                                      (                    t                    )                                                              ⁢                              ⅆ                t                                              )                =                              δ                          m              ,                              m                ′                                              ⁢                      δ                          n              ,                              n                ′                                                                        (        4        )            
One of the prototype functions verifying these conditions is the IOTA prototype function described, for example, in the patent application No. FR 2 733 869, which has the characteristics of being identical to its Fourier transform.
FIG. 1 is a time-frequency representation of the symbols with real values transmitted by OFDM/OQAM modulation and of the symbols with complex values transmitted by classic OFDM modulation without any guard interval.
In this figure, the triangles represent the OFDM/QAM symbols with complex values. The circles and stars for their part represent OFDM/OQAM symbols with real values. For example, the circles correspond to the real part and the stars to the imaginary part of a complex symbol coming from a QAM constellation which it is sought to transmit using a OFDM/OQAM modulation.
Indeed, for a classic OFDM modulation of a complex type, the real and imaginary parts of a complex value coming from the QAM constellation are transmitted simultaneously once at every symbol time Tu; in the case of a real type OFDM/Offset QAM modulation on the contrary, the real and imaginary parts are transmitted with a time lag of half a symbol time (Tu/2).
It is seen in this FIG. 1 that the spectral efficiency of OFDM/OQAM is identical to that of the classic OFDM without guard interval. Indeed, for a same inter-carrier spacing v0, we transmit the following:                in OFDM/OQAM, one real value per carrier at every time interval τ0;        in classic OFDM without guard interval, a complex value (i.e. two real values) every 2*τ0=Tu.        
The quantity of information transmitted by these two modulations is therefore identical. However, the need to introduce a guard interval with a duration Tg in classic OFDM has the effect of reducing the spectral efficiency of classic OFDM as compared with OFDM/OQAM, which proves to be (Tg+2τ0)/2τ0 more efficient.
One drawback of these prior art techniques, whether classic OFDM modulation guard interval or OFDM/OQAM modulation, is that to increase the spectral efficiency it is necessary to use QAM modulations whose constellations have a large number of states. Now, such QAM modulations with large numbers of states are highly sensitive to noise and to errors of estimation of the propagation channel.
Indeed, as indicated here above, the spectral efficiency of the classic OFDM modulation is limited by the need to introduce a guard interval designed to reduce inter-symbol interference during which no payload information can be transmitted. If complex symbols can be transmitted by this OFDM modulation, it is nonetheless true that the associated symbol time is twice as long as it is for an OFDM/OQAM modulation.
OFDM/OQAM modulation for its part, although having a higher spectral efficiency then classic OFDM modulation, is limited by the constraint of orthogonality of the carriers in the real domain, which dictates the choice of modulation filters for the symbols that are well localized in the time-frequency space and which therefore enables the transmission of only symbols with real values.